Partial and Interaction Spline Models for the Semiparametric Estimation of Functions of Several Variables

A partial spline model is a model for a response as a function of several variables, which is the sum of a smooth function of several variables and a parametric function of the same plus possibly some other variables. Partial spline models in one and several variables, with direct and indirect data, with Gaussian errors and as an extension of GLIM to partially penalized GLIM models are described. Application to the modeling of change of regime in several variables is described. Interaction splines are introduced and described and their potential use for modeling non-linear interactions between variables by semiparametric methods is noted. Reference is made to recent work in efficient computational methods

Multivariate Bernoulli distribution

In this paper, we consider the multivariate Bernoulli distribution as a model to estimate the structure of graphs with binary nodes. This distribution is discussed in the framework of the exponential family, and its statistical properties regarding independence of the nodes are demonstrated. Importantly the model can estimate not only the main effects and pairwise interactions among the nodes but also is capable of modeling higher order interactions, allowing for the existence of complex clique effects. We compare the multivariate Bernoulli model with existing graphical inference models - the Ising model and the multivariate Gaussian model, where only the pairwise interactions are considered. On the other hand, the multivariate Bernoulli distribution has an interesting property in that independence and uncorrelatedness of the component random variables are equivalent. Both the marginal and conditional distributions of a subset of variables in the multivariate Bernoulli distribution still follow the multivariate Bernoulli distribution. Furthermore, the multivariate Bernoulli logistic model is developed under generalized linear model theory by utilizing the canonical link function in order to include covariate information on the nodes, edges and cliques. We also consider variable selection techniques such as LASSO in the logistic model to impose sparsity structure on the graph. Finally, we discuss extending the smoothing spline ANOVA approach to the multivariate Bernoulli logistic model to enable estimation of non-linear effects of the predictor variables.Comment: Published in at http://dx.doi.org/10.3150/12-BEJSP10 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

On the optimal choice of nodes in the collocation-projection method for solving linear operator equations

AbstractH R is a Hilbert space with norm ∥ ∥R and K is a linear operator mapping H R into l2(T), where T is a closed, bounded interval of the real line. K and H R jointly have the property that there exist M such that ¦ (Kf)(t)¦ ⩽ M ∥f∥R, t ϵ T. In the collocation-projection method for approximately solving the equation Kf = g, the approximate solution fn is taken as that element of H R that minimizes ∥f;∥R subject to (Kf)(t) = g(t) for t ϵ Tn, where Tn + 1 = {ti}i = 0n and ti ϵ T. We show how results obtained elsewhere are applicable to the problem of choosing the node set Tn to minimize ∥f − fn∥R. The results are applicable to the approximate solution of 2-point boundary value problems using a spline basis. We make some remarks concerning the problem of choosing the nodes to minimize sups ¦f(s) − fn(s)¦, when K is a differential operator

Comment on "Support Vector Machines with Applications"

Comment on "Support Vector Machines with Applications" [math.ST/0612817]Comment: Published at http://dx.doi.org/10.1214/088342306000000457 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Some results on Tchebycheffian spline functions

AbstractThis report derives explicit solutions to problems involving Tchebycheffian spline functions. We use a reproducing kernel Hilbert space which depends on the smoothness criterion, but not on the form of the data, to solve explicitly Hermite-Birkhoff interpolation and smoothing problems. Sard's best approximation to linear functionals and smoothing with respect to linear inequality constraints are also discussed. Some of the results are used to show that spline interpolation and smoothing is equivalent to prediction and filtering on realizations of certain stochastic processes